Mathematical Modeling seminars

Poor air quality in cities is a significant threat to health and life expectancy, with over 80% of people living in urban areas exposed to air quality levels that exceed World Health Organisation limits. In this session, I present a multi-resolution multi-task framework that handles evidence integration under varying spatio-temporal sampling resolution and noise levels. We have developed both shallow Gaussian Process (GP) mixture models and deep GP constructions that naturally handle this evidence integration, as well as biases in the mean. These models underpin our work at the Alan Turing Institute towards providing spatio-temporal forecasts of air pollution across London. We demonstrate the effectiveness of our framework on both synthetic examples and applications on London air quality. For further information go to: https://www.turing.ac.uk/research/research-projects/london-air-quality. Collaborators: Oliver Hamelijnck, Theodoros Damoulas, Kangrui Wang and Mark Girolami.

The Neural Engineering Framework has been used to create a wide variety of biologically realistic brain simulations that are capable of performing simple cognitive tasks (remembering a list, counting, etc.). This includes the largest existing functional brain model. This talk will describe this method, and show some examples of using it to take high-level cognitive algorithms and convert them into a neural network that implements those algorithms. Overall, this approach gives us new ways of thinking about how the brain works and what sorts of algorithms it is capable of performing.

Information that is shared across brains is encoded in idiosyncratic fine-scale functional topographies. Hyperalignment jointly models shared information and idiosyncratic topographies. Pattern vectors for neural responses and connectivities are projected into a common, high-dimensional information space, rather than being aligned in a canonical anatomical space. Hyperalignment calculates individual transformation matrices that preserve the geometry of pairwise dissimilarities between pattern vectors. Individual cortical topographies are modeled as mixtures of overlapping, individual-specific topographic basis functions, rather than as contiguous functional areas. The fundamental property of brain function that is preserved across brains is information content, rather than the functional properties of local features that support that content.

The affinity between statistical physics and machine learning has long history, this is reflected even in the machine learning terminology that is in part adopted from physics. I will describe the main lines of this long-lasting friendship in the context of current theoretical challenges and open questions about deep learning. Theoretical physics often proceeds in terms of solvable synthetic models, I will describe the related line of work on solvable models of simple feed-forward neural networks. I will highlight a path forward to capture the subtle interplay between the structure of the data, the architecture of the network, and the learning algorithm.

The description of neural computations currently relies on two competing views: (i) a classical single-cell view that aims to relate the activity of individual neurons to sensory or behavioural variables, and organize them into functional classes; (ii) a more recent population view that instead characterises computations in terms of collective neural trajectories, and focuses on the dimensionality of these trajectories as animals perform tasks. How the two key concepts of functional cell classes and low-dimensional trajectories interact to shape neural computations is however at present not understood. Here I will address this question by combining machine-learning tools for training recurrent neural networks with reverse-engineering and theoretical analyses of network dynamics.

Cognitive architectures are attempts to build larger-scale models of minds. This talk will explore how structure-mapping models of analogical matching, retrieval, and generalization are used in the Companion cognitive architecture. Examples will include modeling conceptual change, learning by reading, and analogical Q/A training.

Sensory cortices display a suite of ubiquitous dynamical features, such as ongoing noise variability, transient overshoots, and oscillations, that have so far escaped a common, principled theoretical account. We developed a unifying model for these phenomena by training a recurrent excitatory-inhibitory neural circuit model of a visual cortical hypercolumn to perform sampling-based probabilistic inference. The optimized network displayed several key biological properties, including divisive normalization, as well as stimulus-modulated noise variability, inhibition-dominated transients at stimulus onset, and strong gamma oscillations. These dynamical features had distinct functional roles in speeding up inferences and made predictions that we confirmed in novel analyses of awake monkey recordings. Our results suggest that the basic motifs of cortical dynamics emerge as a consequence of the efficient implementation of the same computational function — fast sampling-based inference — and predict further properties of these motifs that can be tested in future experiments

Firing-rate (FR) or neural-mass models are widely used for studying computations performed by neural populations. Despite their success, classical firing-rate models do not capture spike timing effects on the microscopic level such as spike synchronization and are difficult to link to spiking data in experimental recordings. For large neuronal populations, the gap between the spiking neuron dynamics on the microscopic level and coarse-grained FR models on the population level can be bridged by mean-field theory formally valid for infinitely many neurons. It remains however challenging to extend the resulting mean-field models to finite-size populations with biologically realistic neuron numbers per cell type (mesoscopic scale). In this talk, I present a mathematical framework for mesoscopic populations of generalized integrate-and-fire neuron models that accounts for fluctuations caused by the finite number of neurons. To this end, I will introduce the refractory density method for quasi-renewal processes and show how this method can be generalized to finite-size populations. To demonstrate the flexibility of this approach, I will show how synaptic short-term plasticity can be incorporated in the mesoscopic mean-field framework. On the other hand, the framework permits a systematic reduction to low-dimensional FR equations using the eigenfunction method. Our modeling framework enables a re-examination of classical FR models in computational neuroscience under biophysically more realistic conditions.

Complex systems can be modeled at various levels of granularity, e.g., we can model a person at the cognitive level, on the neuronal level, or down to the biochemical level. When multiple models represent the same system at different scales, we would like to be able to reason about the causal effects of interventions on each level in such a way that the models remain consistent across levels. In the first part of this talk, I consider which conditions must be fulfilled for two structural equation models (SEMs) to stand in such a causally consistent relation. In the second part of the talk, I present recent work on learning causally consistent SEMs across multiple levels, distinguishing between bottom-up (micro- to macro-level) and top-down (macro- to micro-level) approaches.

Ideas about optimization are at the core of how we approach biological complexity. Quantitative predictions about biological systems have been successfully derived from first principles in the context of efficient coding, metabolic and transport networks, evolution, reinforcement learning, and decision making, by postulating that a system has evolved to optimize some utility function under biophysical constraints. Yet as normative theories become increasingly high-dimensional and optimal solutions stop being unique, it gets progressively hard to judge whether theoretical predictions are consistent with, or "close to", data. I will illustrate these issues using efficient coding applied to simple neuronal models as well as to a complex and realistic biochemical reaction network. As a solution, we developed a statistical framework which smoothly interpolates between ab initio optimality predictions and Bayesian parameter inference from data, while also permitting statistically rigorous tests of optimality hypotheses.

Theoretical and computational models of neural systems have been traditionally focused on small neural circuits, given the lack of reliable data on large-scale brain structures. The situation has started to change in recent years, with novel recording technologies and large organized efforts to describe the brain at a larger scale. In this talk, Professor Mejias from the University of Amsterdam will review his recent work on developing anatomically constrained computational models of large-scale cortical networks of monkeys, and how this approach can help to answer important questions in large-scale neuroscience. He will focus on three main aspects: (i) the emergence of functional interactions in different frequency regimes, (ii) the role of balance for efficient large-scale communication, and (iii) new paradigms of brain function, such as working memory, in large-scale networks.

Bursts in gamma and other frequency ranges are thought to contribute to the efficiency of working memory or communication tasks. Abnormalities in bursts have also been associated with motor and psychiatric disorders. The determinants of burst generation are not known, specifically how single cell and connectivity parameters influence burst statistics and the corresponding brain states. We first present a generic mathematical model for burst generation in an excitatory-inhibitory (EI) network with self-couplings. The resulting equations for the stochastic phase and envelope of the rhythm’s fluctuations are shown to depend on only two meta-parameters that combine all the network parameters. They allow us to identify different regimes of amplitude excursions, and to highlight the supportive role that network finite-size effects and noisy inputs to the EI network can have. We discuss how burst attributes, such as their durations and peak frequency content, depend on the network parameters. In practice, the problem above follows the a priori challenge of fitting such E-I spiking networks to single neuron or population data. Thus, the second part of the talk will discuss a novel method to fit mesoscale dynamics using single neuron data along with a low-dimensional, and hence statistically tractable, single neuron model. The mesoscopic representation is obtained by approximating a population of neurons as multiple homogeneous ‘pools’ of neurons, and modelling the dynamics of the aggregate population activity within each pool. We derive the likelihood of both single-neuron and connectivity parameters given this activity, which can then be used to either optimize parameters by gradient ascent on the log-likelihood, or to perform Bayesian inference using Markov Chain Monte Carlo (MCMC) sampling. We illustrate this approach using an E-I network of generalized integrate-and-fire neurons for which mesoscopic dynamics have been previously derived. We show that both single-neuron and connectivity parameters can be adequately recovered from simulated data.

During periods of persistent and inescapable stress, animals can switch from active to passive coping strategies to manage effort-expenditure. Such normally adaptive behavioural state transitions can become maladaptive in disorders such as depression. We developed a new class of multi-region recurrent neural network (RNN) models to infer brain-wide interactions driving such maladaptive behaviour. The models were trained to match experimental data across two levels simultaneously: brain-wide neural dynamics from 10-40,000 neurons and the realtime behaviour of the fish. Analysis of the trained RNN models revealed a specific change in inter-area connectivity between the habenula (Hb) and raphe nucleus during the transition into passivity. We then characterized the multi-region neural dynamics underlying this transition. Using the interaction weights derived from the RNN models, we calculated the input currents from different brain regions to each Hb neuron. We then computed neural manifolds spanning these input currents across all Hb neurons to define subspaces within the Hb activity that captured communication with each other brain region independently. At the onset of stress, there was an immediate response within the Hb/raphe subspace alone. However, RNN models identified no early or fast-timescale change in the strengths of interactions between these regions. As the animal lapsed into passivity, the responses within the Hb/raphe subspace decreased, accompanied by a concomitant change in the interactions between the raphe and Hb inferred from the RNN weights. This innovative combination of network modeling and neural dynamics analysis points to dual mechanisms with distinct timescales driving the behavioural state transition: early response to stress is mediated by reshaping the neural dynamics within a preserved network architecture, while long-term state changes correspond to altered connectivity between neural ensembles in distinct brain regions.

Synaptic plasticity is believed to be a key physiological mechanism for learning. It is well-established that it depends on pre and postsynaptic activity. However, models that rely solely on pre and postsynaptic activity for synaptic changes have, to date, not been able to account for learning complex tasks that demand hierarchical networks. Here, we show that if synaptic plasticity is regulated by high-frequency bursts of spikes, then neurons higher in the hierarchy can coordinate the plasticity of lower-level connections. Using simulations and mathematical analyses, we demonstrate that, when paired with short-term synaptic dynamics, regenerative activity in the apical dendrites, and synaptic plasticity in feedback pathways, a burst-dependent learning rule can solve challenging tasks that require deep network architectures. Our results demonstrate that well-known properties of dendrites, synapses, and synaptic plasticity are sufficient to enable sophisticated learning in hierarchical circuits.

Ariel Pakes introduces the foundations of demand estimation for differentiated products. Starting from individual preferences and choices, he explains how consumer heterogeneity can be aggregated into market demand and used to study prices, substitution and welfare. The lecture compares modelling products directly with representing them through characteristics, including observed and unobserved attributes. It examines how different assumptions about tastes generate substitution patterns, using examples involving cars and other differentiated goods. Pakes discusses logit models, vertical and horizontal differentiation, and the implications of demand assumptions for markups, competition and policy analysis.